Since you can cut it in half, this is equivalent to having two ropes of 30 minutes long. Lay them in rows and set fire to opposite ends, so you have this:
Eventually, the flames will meet, like this:
Each is now a 15 minute rope. Set fire to the other end of the smaller rope, so that it burns down in 7.5 minutes. Then set fire to other end of the remaining rope, which will finish burning in 3.75 minutes.
I don't think you can do better than that. I can't think of a rigorous reason, but here's a unrigorous one: Say two ropes have "similar ends" if they burn at the same rate from each end. Given two ropes with different ends and burn time N, the best you can do is N/4. This is because in time N/2, for one rope you can either
- Destroy it (burn both ends)
- Reduce its length by N/2 (burn one end)
You need to destroy at least one rope in order to reduce the rest, because you have no other way of keeping time. So with two ropes you can destroy one to reduce the other to N/2 in N/2 time, then destroy the other in N/4 time.
That's with two ropes with different ends. With two ropes with similar ends, you can also produce two half-length ropes with different ends, so the best you can do is actually N/8. And one rope with both ends similar can be cut in half to produce two ropes with similar ends, so you can do N/16.
The bold part is the handwavey part, because I can't think of a strong reason why there isn't a more creative, better thing you can do there.